A372326 - OEIS

A372326

Number A(n,k) of acyclic orientations of the Turán graph T(k*n,n); square array A(n,k), n>=0, k>=1, read by antidiagonals.

%I #29 Jun 09 2024 09:02:22

%S 1,1,1,1,1,1,1,1,2,1,1,1,14,6,1,1,1,230,426,24,1,1,1,6902,122190,

%T 24024,120,1,1,1,329462,90768378,165392664,2170680,720,1,1,1,22934774,

%U 138779942046,4154515368024,457907248920,287250480,5040,1

%N Number A(n,k) of acyclic orientations of the Turán graph T(k*n,n); square array A(n,k), n>=0, k>=1, read by antidiagonals.

%C The Turán graph T(k*n,n) is the complete n-partite graph K_{k,...,k}.

%C An acyclic orientation is an assignment of a direction to each edge such that no cycle in the graph is consistently oriented. Stanley showed that the number of acyclic orientations of a graph G is equal to the absolute value of the chromatic polynomial X_G(q) evaluated at q=-1.

%H Alois P. Heinz, <a href="/A372326/b372326.txt">Antidiagonals n = 0..42, flattened</a>

%H Richard P. Stanley, <a href="http://dx.doi.org/10.1016/0012-365X(73)90108-8">Acyclic Orientations of Graphs</a>, Discrete Mathematics, 5 (1973), pages 171-178, doi:10.1016/0012-365X(73)90108-8

%H Wikipedia, <a href="https://en.wikipedia.org/wiki/Acyclic_orientation">Acyclic orientation</a>

%H Wikipedia, <a href="https://en.wikipedia.org/wiki/Multipartite_graph">Multipartite graph</a>

%H Wikipedia, <a href="https://en.wikipedia.org/wiki/Tur%C3%A1n_graph">Turán graph</a>

%F A(n,k) = A267383(k*n,n).

%e Square array A(n,k) begins:

%e 1, 1, 1, 1, 1, ...

%e 1, 2, 14, 230, 6902, ...

%e 1, 6, 426, 122190, 90768378, ...

%e 1, 24, 24024, 165392664, 4154515368024, ...

%e 1, 120, 2170680, 457907248920, 495810323060597880, ...

%p g:= proc(n) option remember; `if`(n=0, 1, add(

%p expand(x*g(n-j))*binomial(n-1, j-1), j=1..n))

%p end:

%p A:= proc(n, k) option remember; local q, l, b; q, l, b:= -1, [k$n, 0],

%p proc(n, j) option remember; `if`(j=1, mul(q-i, i=0..n-1)*

%p (q-n)^l[1], add(b(n+m, j-1)*coeff(g(l[j]), x, m), m=0..l[j]))

%p end; abs(b(0, nops(l)))

%p end:

%p seq(seq(A(n, d-n), n=0..d), d=0..10);

%t g[n_] := g[n] = If[n == 0, 1, Sum[Expand[x*g[n - j]]*Binomial[n - 1, j - 1], {j, 1, n}]];

%t A[n_, k_] := A[n, k] = Module[{q = -1, l, b}, l = Append[Table[k, {n}], 0];

%t b[nn_, j_] := b[nn, j] = If[j == 1, Product[q - i, {i, 0, nn - 1}]*

%t (q - nn)^l[[1]], Sum[b[nn + m, j - 1]*Coefficient[g[l[[j]]], x, m],

%t {m, 0, l[[j]]}]];

%t Abs[b[0, Length[l]]]];

%t Table[Table[A[n, d - n], {n, 0, d}], {d, 0, 10}] // Flatten (* _Jean-François Alcover_, Jun 09 2024, after _Alois P. Heinz_ *)

%Y Columns k=0-2 give: A000012, A000142, A033815.

%Y Rows n=0+1,2-3 give: A000012, A048163(k+1), A370961.

%Y Main diagonal gives A372084.

%Y Cf. A267383.

%K nonn,tabl

%O 0,9

%A _Alois P. Heinz_, Apr 27 2024